We prove that the leaves of an inverse mean curvature flow provide a foliation of a future end of a cosmological spacetime $N$ under the necessary and sufficient assumptions that $N$ satisfies a future mean curvature barrier condition and a strong volume decay condition. Moreover, the flow parameter t can be used to define a new physically important time function.

This paper is the continuation of Causal properties of AdS-isometry groups I: causal actions and limit sets. We essentially prove that the family of strongly causal spacetimes defined in Causal properties of AdS-isometry groups I: causal actions and limit sets associated to generic achronal subsets in Ein₂ contains all the examples of BTZ multi-blackholes. It provides new elements for the global description of these multiblack-holes. We also prove that any strongly causal spacetime locally modeled on the anti-de Sitter space admits a well-defined maximal strongly causal conformal boundary locally modeled on Ein₂.

We construct topological string and topological membrane actions with a nontrivial 3-form flux $H$ in arbitrary dimensions. These models realize Bianchi identities with a nontrivial $H$ flux as consistency conditions. Especially, we discuss the models with a generalized $SU(3)$ structure, a generalized G₂ structure and a generalized Spin(7) structure. These models are constructed from the AKSZ formulation of the Batalin–Vilkovisky formalism.

We study the sewing constraints for rational two-dimensional conformal field theory on oriented surfaces with possibly nonempty boundary. The boundary condition is taken to be the same on all segments of the boundary. The following uniqueness result is established: for a solution to the sewing constraints with nondegenerate closed state vacuum and nondegenerate two-point correlators of boundary fields on the disk and of bulk fields on the sphere, up to equivalence all correlators are uniquely determined by the one-, two- and three-point correlators on the disk.

Thus for any such theory every consistent collection of correlators can be obtained by the topological field theory approach of our papers TFT construction of RCFT correlators I: partition functions and TFT construction of RCFT correlators V: Proof of modular invariance and factorisation. As morphisms of the category of world sheets we include not only homeomorphisms, but also sewings; interpreting the correlators as a natural transformation then encodes covariance both under homeomorphisms and under sewings of world sheets.

For a one-dimensional discrete Schrödinger operator with a weakly coupled potential given by a strongly mixing dynamical system with power law decay of correlations, we derive for all energies including the band edges and the band center a perturbative formula for the Lyapunov exponent. Under adequate hypothesis, this shows that the Lyapunov exponent is positive on the whole spectrum. This in turn implies that the Hausdorff dimension of the spectral measure is zero and that the associated quantum dynamics grows at most logarithmically in time.

We study cohomological gauge theories on total spaces of holomorphic line bundles over complex manifolds and obtain their reduction to the base manifold by $U(1)$-equivariant localization of the path integral. We exemplify this general mechanism by proving via exact path integral localization a reduction for local curves conjectured in hep-th/0411280, relevant to the calculation of black hole entropy/Gromov–Witten invariants. Agreement with the four-dimensional gauge theory is recovered by taking into account in the latter non-trivial contributions coming from one-loop fluctuation determinants at the boundary of the total space. We also study a class of abelian gauge theories on Calabi–Yau local surfaces, describing the quantum foam for the $A$-model, relevant to the calculation of Donaldson–Thomas invariants.